[To RWG: My naive look would have yielded a degree estimate a good deal greater than 8, but the degree reductions are probably sort of low-dimensional miracles that eventually in high dimensions will not have much effect?] Suppose we begin with two unit hypercubes, one is [-1/2,1/2]^D, other is same but translated 10 along the x-axis. Now, rotate the first hypercube in the (x,x1) plane by angle 1/9 radian, in the (x,x2) plane by angle 1/9^2, in the (x,x3) plane by angle 1/9^3, etc. Here x1,x2,x3,... all are axes orthogonal to each other and to the x-axis. Now claim the second hypercube will, by moving it back down the x-axis, pass thru the first hypercube, because those rotations "widened" it in all directions orthogonal to x-axis. If that argument is valid, then it shows the first D-dimensional hypercube can actually be made slightly smaller than unit by about 1 part in 9^(D-1), and a unit hypercube still will pass thru it. Better would be to figure out what the max-allowed rotation angles are, which would not necessarily be optimum, but would provide a sort of greedy-like crude guess at the optimum, and which would be a valid lower bound. As an upper bound, the scaling factor must be less than squareroot(D/(D-1)) by considering diameters of the D-hypercube and (D-1)-hypercube. So there is a pretty large gap between this upper bound, and the putative lower bound (which I have not actually worked out, I merely sketched roughly a way one might try to do it) -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)